A person draws . two cards with replacement from a pack of 52 cards . What is the chance that he gets both cards of the same suit?

Options

A. `(1)/(4)`
B. `(3)/(13)`
C. `(1)/(16)`
D. `(2)/(13)`

Answer

To solve the problem of finding the probability that a person draws two cards of the same suit from a pack of 52 cards with replacement, we can follow these steps:

### Step 1: Understand the total number of outcomes
When drawing two cards with replacement, the total number of ways to draw the first card is 52 (since there are 52 cards in total). After replacing the first card, the second card can also be any of the 52 cards. Therefore, the total number of outcomes when drawing two cards is:

\[ \text{Total outcomes} = 52 \times 52 = 2704 \]

**Hint:** When calculating total outcomes with replacement, multiply the number of options for each draw.

### Step 2: Determine the favorable outcomes
Next, we need to find the number of favorable outcomes where both cards drawn are of the same suit. There are 4 suits in a deck of cards (hearts, diamonds, clubs, spades), and each suit has 13 cards.

1. Choose a suit: There are 4 ways to choose a suit.
2. Choose the first card from that suit: There are 13 options.
3. Choose the second card from the same suit: Again, there are 13 options.

Thus, the number of favorable outcomes is calculated as:

\[ \text{Favorable outcomes} = 4 \times 13 \times 13 = 4 \times 169 = 676 \]

**Hint:** To find favorable outcomes, consider the number of ways to choose both the suit and the cards from that suit.

### Step 3: Calculate the probability
Now that we have both the total outcomes and the favorable outcomes, we can calculate the probability of drawing two cards of the same suit using the formula:

\[ \text{Probability} = \frac{\text{Favorable outcomes}}{\text{Total outcomes}} = \frac{676}{2704} \]

### Step 4: Simplify the probability
To simplify \( \frac{676}{2704} \):

1. Divide both the numerator and denominator by 676:

\[ \frac{676 \div 676}{2704 \div 676} = \frac{1}{4} \]

Thus, the probability that both cards drawn are of the same suit is:

\[ \text{Probability} = \frac{1}{4} \]

### Final Answer
The probability that a person draws two cards of the same suit is \( \frac{1}{4} \).

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